Write an equation of area A of the enclosure as a function of x for the rectangular area.

Question

Three sides of a fence and an existing wall form a rectangular enclosure. The total length of a fence used for the three sides is 240 feet. Let x be the length of two sides perpendicular to the wall. Write an equation of area A of the enclosure as a function of x for the rectangular area.

Answer 1

let y=length and let x=width (the perpendicular sides)

P=2y+2x, but the wall is being used as one of the lengths, so...

P=y+2x

240=y+2x

240-2x=y

A=xy

A=x(240-2x)

A=240x-2x²

if you want the maximum area, use -b/2a

-b/2a=-240/-4=60 feet for each width and 240-60-60=120 feet for the one length

A=60*120=7200 square feet

Answer 2

Let y be the length of the side parallel to the wall

 

Then, 2x + y = 240.  So, y = 240 - 2x

 

Area = xy = x(240 - 2x) = 240x - 2x²  

 

Note that 0 < x < 120  (x can't be negative or 0 because it is a length and if x ≥ 120, then y would be ≤ 0, which doesn't make sense either)

Answer 3

Draw and label of diagram.

 

x represents the length of the two sides perpendicular to the wall.

240 - 2x represents the length of the side parallel to the wall

 

A = x(240 - 2x)

A = 240x - 2x²